Classes of Samuel ORSO
Class	Time
PTDS	8h30 - 12h00

Classes of Aleksandr SHEMENDYUK
Class	Time
PTDS	8h30 - 12h00

R Markdown Syntax

Cache example

Example of a situation where the cache = T leads to incorrect calculations:

set.seed(123)
randomNumber <- rnorm(1)
randomNumber

## [1] -0.5604756

This chunk will store a randomly generated number. However, because we set cache = T, when we rerun the chunk, the number won’t change as it will be retrieving the number from the cache.

randomNumber

## [1] -0.5604756

In the chunk above, if we need to get a different number, we will get the same number every time we run it because we used cache = T in the first chunk. This might not be beneficial especially in situations where we require different outputs every time we run the code.

Random sample

set.seed(123)
# Generate 2000 random samples
lambda <- 2
y <- rpois(2000, lambda)

# Compute empirical mean and variance
y_mean <- mean(y)
y_var <- var(y)

# Theoretical mean and variance of a Poisson distribution
theoretical_mean <- 2
theoretical_variance <- 2

# Compare
print(paste("Empirical mean: ", y_mean))

## [1] "Empirical mean:  1.9915"

print(paste("Theoretical mean: ", theoretical_mean))

## [1] "Theoretical mean:  2"

print(paste("Empirical variance: ", y_var))

## [1] "Empirical variance:  1.97441495747874"

print(paste("Theoretical variance: ", theoretical_variance))

## [1] "Theoretical variance:  2"

The empirical mean and variance should be close to the theoretical value ($\lambda = 2$) but might not match precisely. This is because of the randomness inherent in the simulation. With larger sample sizes, the empirical values should get closer to the theoretical values due to the Law of Large Numbers.

Difference between the theoretical and observed values for lambda = 2.
Measure	Sample estimate	Theoretical	Absolute difference
Mean	1.992	2	0.008
Variance	1.974	2	0.026

To display the histogram of $y$, one would use the following command:

hist(y, breaks = 10, xlab = "Data values", ylab = "Frequency")

Historgram of y with 10 bins

adding as an option to this chunk the option fig.cap="Historgram of y with 10 bins" to display the caption to this figure. Here, we do not observe the accurate shape of the Poisson distribution with $\lambda = 2$, which should be more centred around the value of 2.

hist(y, breaks = 50, xlab = "Data values", ylab = "Frequency")

Historgram of y with 50 bins

Increasing the number of bins to 50 results in a more granular view of the data distribution. However, it could also lead to overfitting where too much noise is mistaken for an actual pattern. It’s important to choose an appropriate number of bins to accurately convey the distribution of the data in a histogram.

Equations

Include the following equation

\[ f(x;\, \mu, \sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2 / (2\sigma ^2)} \]

This formula represents the Probability Density Function of a Normal Distribution where $\mu$ is the mean and $\sigma^2$ is the variance.

Include the following in-line equation: eq2

$ \hat{p}_{stress} =k s^{-m}$

This is a typical Stress-Life equation in material science, where $\hat{p}_{stress}$ represents the predicted stress, $s$ is the life of the material, and $k$ and $m$ are material constants.

On how to include references in Markdown, see the guide.

As shown in Choi, Marstaller, and Aceves (2023), the singular integral representation of the fractional Laplacian is defined as

\[ (-\Delta)^{\frac{\alpha}{2}} u(x)=c(d, \alpha) \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y:=\frac{2^\alpha \Gamma\left(\frac{\alpha}{2}+\frac{d}{2}\right)}{\pi^{\frac{d}{2}}\left|\Gamma\left(-\frac{\alpha}{2}\right)\right|} \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y . \]

\[ (-\Delta)^{\frac{\alpha}{2}} u(x)=c(d, \alpha) \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y:=\frac{2^\alpha \Gamma\left(\frac{\alpha}{2}+\frac{d}{2}\right)}{\pi^{\frac{d}{2}}\left|\Gamma\left(-\frac{\alpha}{2}\right)\right|} \text { p.v. } \int_{\mathbb{R}^d} \frac{u(x)-u(y)}{|x-y|^{d+\alpha}} d y . \]

Text

<span style="color:blue">"The only thing that is constant is change.", Heraclitus.</span>

“The only thing that is constant is change.”, Heraclitus.

Button

<button data-toggle="collapse" data-target="#demo">Show/Hide Answer</button> <div id="demo" class="collapse"> This is the solution to the problem. </div>

This is the solution to the problem.

Color box

<div style="background-color: #F5F5DC; padding: 10px; border: 1px solid black;"> "Learning never exhausts the mind." - Leonardo da Vinci </div>

“Learning never exhausts the mind.” - Leonardo da Vinci

Homework 1 - Solution

Samuel Orso

Aleksandr Shemendyuk

2023-10-19

Introduction

Group Members

Samuel ORSO

Aleksandr SHEMENDYUK